Ducted and Contrarotating Propellers on Merchant Ships 



radius, thereby assuming optimum circulation distribution. These values were 

 used by Morgan. In our scheme Lerbs' original values have been used, as the 

 improvement obtained by using the values of Tachmindji seemed to be doubtful 

 for the circulation distributions normally used in our calculations. Instead, a 

 modified version of the computer program, which is now under work, will con- 

 tain a calculation of these factors for arbitrary circulation distributions. 



(4) The calculations are carried out for corresponding radii for the 

 forward and aft propellers as defined by Lerbs: 



where r ^ is the local radius of the forward propeller, r ^ is the corresponding 

 local radius for aft propeller, and s^ is the contraction of streamline. 



For calculating the contraction s ^ Lerbs applied the equation of continuity 

 to each annular element. By introducing Eq. (2) and neglecting second-order 

 terms, a linear differential equation was obtained for s^, the solution of which 

 is a definite integral (see (9) or (10)). According to the authors' experience, 

 this way of calculating the contraction is not accurate enough and in our scheme 

 the contraction is obtained by direct numerical integration. Thereby no simpli- 

 fications are necessary. The diameter D^ of the aft propeller is determined, as 

 by Lerbs and Morgan, by the relation 



D^ = Z)i(i-Si) , (2a) 



where D^ is the diameter of forward propeller and Sj = s^ at the blade tips. 



(5) What has been said above applies to the lifting-line calculations. 

 After the completion of these calculations lifting -surface calculations are car- 

 ried out according to a method based on Pien's approach (13, 14), giving correc- 

 tions on camber and pitch. Finally, an approximate correction of the pitch for 

 thickness effect is added (14). In the lifting-surface calculations and the calcu- 

 lations of the thickness effect, the mutual interference of the propellers is 

 neglected. 



3.2. Systematic Series of Contrarotating Propellers 



For checking the design method and providing figures of the possible effi- 

 ciencies, a systematic series consisting of four sets of contrarotating propellers 

 was designed, manufactured, and tested in open-water condition in the SSPA 

 cavitation tunnel. The propellers were fitted with adjustable blades. Some of 

 the important design data of the propellers are given in Table 1, together with 

 the pitch ratios investigated with the different sets. Further, the blade form, 

 pitch distributions, and radial distributions of maximum camber of the blade 

 sections of the propellers are shown in Figs. 4-6. The results of the open- 

 water tests with the four propeller sets at the design pitch are compared with 

 the computed propeller characteristics for the design point in Table 2. In Fig. 7 

 the difference of number of revs, between calculation and experiment are shown 

 for the four propeller sets, together with the corresponding results obtained 



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